Yes, you can get a DUI on a horse. If you use anything besides your own feet to travel around while drunk, it's a DUI. And if you're drunk and walking down the street, it's a drunk in public. You're fucked either way.

You cannot get a DUI or an OUI on anything considered to be a beast of burden. Nor can you get either on roller skates, a skateboard, or similar device. However, you can get a public intoxication at any point, even if you're sober.